This section develops bounds on \(|s_n(z) - \bar{s}_n(z)|\)  and \(|\frac{d}{dz}(s_n(z) - \bar{s}_n(z))|\)  based on the work of Zhidong Bai, Jack Silverstein, Jiang Hu, and Wang Zhou, in particular, Section 4.1 of [1]} and Section 8 of [2]}. The bounds are novel in that (1) dependence on \(\beta _n\)  is more carefully tracked and (2) the bound applies to \(z \ge 1 + (2+\eta ) \sqrt{\beta _n}\)  (rather than to \(z\)  with \(\Im (z) > 1/\sqrt{m}\) ). (2) is critical since the proof of Theorem REF  requires bounding \(|s_n(z) - \bar{s}_n(z)|\)  and \(|\frac{d}{dz} (s_n(z) - \bar{s}_n(z))|\)  along a contour centered at a real point.
